Problem: All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$3.50$ each for students, and the group paid $$54.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$30.00$ each for teachers and $$10.00$ each for students, and the group paid $$180.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+3.5y = 54}$ ${30x+10y = 180}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-14y = -216}$ ${30x+10y = 180}$ Add the top and bottom equations together. $ -4y = -36 $ $ y = \dfrac{-36}{-4}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {7.5x+3.5y = 54}$ to find $x$ ${7.5x + 3.5}{(9)}{= 54}$ $7.5x+31.5 = 54$ $7.5x = 22.5$ $x = \dfrac{22.5}{7.5}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {30x+10y = 180}$ and get the same answer for $x$ ${30x + 10}{(9)}{= 180}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.